F Test Statistic: Step-by-Step Calculation

Updated 11 November 2026 · Standard formula, shown below

This guide explains how to calculate the F test statistic, a ratio used in ANOVA and regression analysis to compare variances or test whether group means are significantly different. You’ll need two sets of data (or two variance estimates), and you’ll get a single F value that you can compare against a critical value from an F-distribution table.

What You’ll Need

Two samples or two variance estimates (e.g., between-group and within-group variances in ANOVA).
Degrees of freedom for each variance (df1 for numerator, df2 for denominator).
A calculator or spreadsheet for basic arithmetic.
An F-distribution table (or software) to interpret the result.

Step-by-Step

Identify the two variances. In a simple F-test (e.g., comparing two population variances), label the larger sample variance as s1² (numerator) and the smaller as s2² (denominator). In ANOVA, the numerator is the mean square between groups (MSB) and the denominator is the mean square within groups (MSW).
Calculate the variance for each group (if not given). For each sample, compute the mean, then sum the squared deviations from the mean, and divide by (n-1) for sample variance. Formula: s² = Σ(xi – x̄)² / (n-1).
Determine the degrees of freedom. For each variance, df = n – 1 (for a single sample). In ANOVA, df1 = k – 1 (k = number of groups) and df2 = N – k (N = total observations).
Divide the numerator variance by the denominator variance. F = s1² / s2² (or MSB / MSW). Ensure the numerator is the larger variance (or the one you’re testing).
Record the F statistic and both degrees of freedom. You now have F(df1, df2) = value.
Compare with the critical value. Look up the critical F in a table using your chosen alpha (e.g., 0.05) and the two degrees of freedom. If your F is greater than the critical value, the result is statistically significant.

Common Mistakes

Putting the smaller variance in the numerator, which flips the F value and can lead to wrong conclusions.
Using the wrong degrees of freedom (e.g., using n instead of n-1).
Confusing sample variance (divides by n-1) with population variance (divides by n).
Forgetting to square standard deviations if you have them; F requires variances, not standard deviations.
Using the wrong denominator in ANOVA (e.g., using total mean square instead of within-group mean square).

Tips

Always label which variance is the numerator and which is the denominator in your work.
Use a spreadsheet (like Excel) to compute variances: =VAR.S() for sample variance.
For ANOVA, many software packages output the F statistic directly; you just need to read the ANOVA table.
When in doubt, check that your F value is positive (it always is) and that df1 is the numerator’s df.

Troubleshooting

F value is negative or zero? Variances are always non-negative, so this indicates a calculation error. Recheck your squared deviations and division.
F value seems extremely large or small? Verify that you divided the correct variances and that you used the right degrees of freedom. Also check if you accidentally used standard deviations instead of variances.
Can’t find the critical value in the table? Most tables list df1 across the top and df2 down the side. If your df is not listed, use the nearest smaller value (conservative approach) or use software.
Not sure which variance goes on top? For a two-tailed test, put the larger variance on top. For a one-tailed test, follow your hypothesis (the variance you expect to be larger).

Frequently Asked Questions

What does the F statistic measure?

It measures the ratio of two variances. A value near 1 suggests the variances are similar; a large value suggests they are different.

What is the difference between F-test and ANOVA?

ANOVA uses the F-test to compare means across multiple groups. The F statistic in ANOVA is the ratio of between-group variance to within-group variance.

Can the F statistic be less than 1?

Yes, if the denominator variance is larger than the numerator. However, in a two-tailed test, we often put the larger variance on top to get F ≥ 1.

What are degrees of freedom in an F-test?

They are the sample sizes minus one for each variance (or k-1 and N-k for ANOVA). They determine the shape of the F-distribution and the critical value.

How do I interpret the p-value from an F-test?

If the p-value is less than your significance level (e.g., 0.05), you reject the null hypothesis that the variances (or means) are equal.

What if my data is not normally distributed?

The F-test is sensitive to non-normality. Consider using Levene’s test or Brown-Forsythe test for equality of variances.

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.